Sunday, August 12, 2012

So I've been revising my notes on general relativity, and I've found several things worth mentioning.

1. Equivalence Principle. The equivalence principle gives us geometry. This is often poorly described (I too committed this error in my drafts).

The equivalence principle tells us neither the composition of a body nor its mass determines its trajectory in a gravitational field. So gravity determines paths, and this gives us geometry.

Moreover, there are different equivalence principles which should be mentioned. I yielded to this, and became incoherent (alas!). The trick is to stick this into a box, for the interested reader to find out more about it, but not obstruct the writing.

Normally this is of the form "ad(u)v∈Lie(G)" and is shorthand for "ad(u)=[u,-]".

The Jacobi identity looks like:

ad(u)ad(v)-ad(v)ad(u)=ad(ad(u)v).

This is the most important identity. Vertex operator algebras are an algebra with a similar property.

A vertex operator algebra consists of a vector space V equipped with a mapping usually denoted

Y:V→(End V)[[x,x-1]].

In this form, it looks like left-multiplication operator...or that's the intuition anyways. So if "v∈V", we should think Y(v,x) belongs to "(End V)[[x,x-1]]" and acts on the left.

Really through currying this should be thought of as "V⊗V→V[[x,x-1]]", i.e., a sort of multiplication operator with a parameter "x". (This is related to the "state-operator correspondence" physicists speak of with conformal field theories.)

Just like a Lie algebra, the Vertex Operator algebra satisfies a Jacobi identity and it is the most important defining property for the VOA.

Desired Property: for any Laurent polynomial f(z) we have δ(z-1)f(z)=δ(z-1)f(1).

This turns out to be true, thanks to the magic of infinite series; but due to html formatting, I omit the proof. The proof is left as an exercise to the reader (the basic sketch is consider δ(z-1)zn, then prove linearity, and you're done).